 So, we understand a lot about the spectroscopy for diatomic molecules, particularly once we include all of the various corrections that we understand at this point. What can we understand about polyatomic molecules? For example, CO2. If I put up on the screen here a spectrum for the absorption of light by CO2 in the infrared portion of the spectrum, we see that the spectrum looks like this. And in particular, let's focus in on these frequencies a little above 2,300 wave numbers. If I zoom in on this region of the spectrum, so this portion, this inset is just showing us a zoomed in portion at 22 or 2,300 wave numbers. So we can see that the structure of this spectrum looks an awful lot like it did for the diatomic molecules that we've talked about so far. There's a fundamental vibrational frequency, some additional rotation or some reduced rotational energy as we excite row vibrational energy levels. There's even these smaller peaks that are redshifted a little further that represent the n equals 1 to n equals 2 increase. So those look an awful lot like the row vibrational peaks that we've seen for diatomic molecules. The spectrum also includes this feature down here, which is again a pair of different lobes surrounding a fundamental vibrational frequency. So it looks like the excitation, the frequencies that are absorbed by this triatomic molecule have the same structure as they do for a diatomic molecule. And the reason for that is a triatomic molecule doesn't do anything qualitatively any different than a diatomic molecule does, it just may have a different number of vibrations or different number of rotations. If we think back to what we know about degrees of freedom, we can calculate exactly how many different rotations and vibrations we should expect the molecule to have. So if we count degrees of freedom, there might be translational motions or rotational motions or vibrational motions in total since it's a triatomic molecule with three atoms, we expect nine total degrees of freedom. As with any molecule, three of them are going to be translations, it can move in x, y, or z. It's a linear molecule, so we have to know the structure of CO2. CO2 has a Lewis structure that looks like this. So with only two substituents around the center carbon and no lone pairs, that's a linear molecule. So the linear molecules can rotate in only two different ways. As a reminder, I could rotate the molecule in this direction or I can rotate it out of the board. But if I spin it along this axis, it's not changing the molecule. So there's only two rotations that change the molecule. So what that means is these five need to be added to four different vibrations in order to sum up to nine total degrees of freedom. So what we've just learned is there's two rotations and four distinct vibrational motions for this molecule. And in fact, those vibrations are well understood. The different vibrations that the CO2 molecule can undergo, it has a symmetric stretch vibration. I'll draw little cartoons of each of these motions. So for the symmetric stretch vibration, both of the CO bond lengths get longer at the same time. And then after they get long enough, they turn around and come back and then they both get shorter at the same time. So the CO stretches are happening symmetrically. There's also an asymmetric stretch. And that is when one of the bonds is getting longer at the same time, another one is getting shorter. So this CO bond on the right is getting longer while the other oxidant is moving toward the central carbon. Here I need an extra caveat because we need these to be vibrational motions. They're not rotations and they're not translations. So that means the center of the mass of the molecule is not moving and the orientation of the molecule is not changing. If I just do what I've shown, if I move these two oxygen atoms to the right, and the carbon doesn't move, then the center of the mass of the molecule has moved to the right somewhat. This bond has gotten shorter and that bond has gotten longer, but the center of the molecule has moved to the right. So to prevent it from having some translational character, I need the carbon itself to be moving to the left. So if I make the carbon move to the left by the proper amount to keep the center of mass stationary, then I've got a lengthened CO bond over here and a shortened CO bond over there. That's a more correct description of this asymmetric stretch motion of the molecule. There's two additional vibrations. There's a symmetric stretch, an anti-symmetric stretch. There's a bending mode that involves the bond angle changing from 180 degrees to smaller values. So if the oxygen's move upward, that's going to decrease the bond angle at the carbon. But again, in order to prevent that from having some translational character, the carbon needs to move down at the same time. So that's the description of a bending motion of the molecule. There's in fact two different bending motions. If I take this picture and I rotate it 90 degrees until it's out of the plane of the light board, then if this oxygen is moving towards you, and this oxygen is also moving towards you, but this carbon is moving away from you, that would be a bending motion where the oxygen is moving in the horizontal plane rather than in the vertical plane. So those are two different bending motions that the CO2 molecule can undergo. So those altogether are the four different vibrational molecules that the CO2 molecule exhibits. And we know there aren't any more because we can count the number of vibrations. If I tell you the vibrational frequencies, each one of these vibrations has its own characteristic frequency. So the vibrational frequency associated with the symmetric stretch is 1340 wave numbers, asymmetric stretch is 2350 wave numbers or so. And the two different bends, as you might guess, are degenerate. They have the same frequency as each other because they're the same motion just in different directions. So now that we know there's four vibrational frequencies, it makes sense that there's no new types of peaks in this row vibrational spectrum. There's still row vibrational peaks where the molecule changes one of these vibrations and either tax on some additional rotational energy or loses a little bit of rotational energy. So the structure of these peaks is very familiar. We might ask, though, why do we only see two of these? We see one over here, one over there. There's four different vibrational frequencies. Why do we not see four of these distinct row vibrational bands at four different frequencies? One answer is because these two are degenerate. This peak that is centered near 667 wave numbers, the excitation energies are identical for this CO2 bending motion and this CO2 bending motion. So those two are layered directly on top of each other. 2349, that's this band right here. So 2349 plus a little bit, minus a little bit, are the row vibrational energy for transitions involving the asymmetric stretch plus or minus some amounts of rotational energy. The only mystery then is where is this band? Why is there not a sequence of absorptions at 1,340 wave numbers? Why do we not see absorption here? And to answer that question, we need to remember what the selection rules are. In particular, the gross selection rules are for vibrational excitations. Recall that we need for it to be true that the dipole moment, as I change the value of the bond displacement, as I cause the bond to be stretched and compressed, I need for that to be non-zero in order for a molecule to absorb light and change its vibrational state. So let's think about the dipole moment of the CO2 molecule. Oxygens are more electronegative than carbon, so I've got negative charges on the two ends of the molecule and positive charge in the middle. With those 180 degree bond angles, the bond dipole for this CO bond exactly cancels the bond dipole for that CO bond, and the dipole moment of this molecule is zero. So that might sound a little worrisome. How can I obey the selection rule if I have a dipole moment of zero? Here's where it's important to remember, we don't need the dipole moment to be non-zero. We need the dipole moment to change as the vibrational motion occurs. So for several of the motions, like the bending motion, for example, if I take these two negative charges and I move them upward, move the carbon downward, that is going to develop a dipole moment. The dipole moment changes from zero to some non-zero value as the molecule bends. So these two motions are, in fact, infrared active, which is why we see absorption at 667 wave numbers plus or minus some rotational amounts. Likewise for the asymmetric stretch, if this CO bond is getting longer, this CO bond is getting shorter, then the two bond dipoles don't cancel each other anymore. And the molecule originally with no dipole moment develops a dipole moment as the asymmetric stretch changes the bond lengths. So this vibrational mode is infrared active. The reason we don't see any absorption at 1340 wave numbers is that the symmetric stretch is infrared inactive. If the two bonds stretch by an equal amount, then it remains true that the bond dipole for the right hand, CO bond, and the dipole moment for the left hand, CO bond, remain exactly equal to each other but pointing in opposite directions so they cancel. So for the symmetric stretch, the dipole moment remains zero as the vibration takes place, and that violates the selection rule, causing this particular vibrational mode to be infrared inactive. So we don't see any absorption at 1340 wave numbers. So basic point we've illustrated is that the vibrational spectroscopy in the infrared portion of the spectrum for a polyatomic molecule like CO2 is qualitatively the same as for a diatomic molecule, but instead of one band of frequencies where absorption occurs, we might get more than one. The number of bands where we see absorption depends on these vibrational frequencies, and whether the modes are infrared active or inactive. So as we see molecules with more distinct vibrational modes, there will be more and more bands appearing in the spectrum. And so we'll take a look at another example next of a molecule with several different bands in its infrared spectrum.